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Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets).
In mathematics, the Chambolle-Pock algorithm is an algorithm used to solve convex optimization problems. It was introduced by Antonin Chambolle and Thomas Pock [1] in 2011 and has since become a widely used method in various fields, including image processing, [2] [3] [4] computer vision, [5] and signal processing.
An interior point method was discovered by Soviet mathematician I. I. Dikin in 1967. [1] The method was reinvented in the U.S. in the mid-1980s. In 1984, Narendra Karmarkar developed a method for linear programming called Karmarkar's algorithm, [2] which runs in provably polynomial time (() operations on L-bit numbers, where n is the number of variables and constants), and is also very ...
In fact, while any closed convex semialgebraic set in the plane can be written as a feasible region of a SOCP, [8] it is known that there exist convex semialgebraic sets that are not representable by SDPs, that is, there exist convex semialgebraic sets that can not be written as a feasible region of a SDP. [9]
In some linear optimization problems, even though the number of constraints is exponential, one can still write a custom separation oracle that works in polynomial time. Some examples are: The minimum-cost arborescence problem: given a weighted directed graph and a vertex r in it, find a subgraph of minimum cost that contains a directed path ...
MOSEK is a software package for the solution of linear, mixed-integer linear, quadratic, mixed-integer quadratic, quadratically constrained, conic and convex nonlinear mathematical optimization problems. The applicability of the solver varies widely and is commonly used for solving problems in areas such as engineering, finance and computer ...
The Frank–Wolfe algorithm is an iterative first-order optimization algorithm for constrained convex optimization.Also known as the conditional gradient method, [1] reduced gradient algorithm and the convex combination algorithm, the method was originally proposed by Marguerite Frank and Philip Wolfe in 1956. [2]
Consider a family of convex optimization problems of the form: minimize f(x) s.t. x is in G, where f is a convex function and G is a convex set (a subset of an Euclidean space R n). Each problem p in the family is represented by a data-vector Data( p ), e.g., the real-valued coefficients in matrices and vectors representing the function f and ...